Let P be an arbitrary point having sum of the squares of the distances from the planes x + y + z = 0, lx $-$ nz = 0 and x $-$ 2y + z = 0, equal to 9. If the locus of the point P is x² + y² + z² = 9, then the value of l $-$ n is equal to _________.

Let point P is ($\alpha$, $\beta$, $\gamma$)<br><br>$${\left( {{{\alpha + \beta + \gamma } \over {\sqrt 3 }}} \right)^2} + {\left( {{{l\alpha - n\gamma } \over {\sqrt {{l^2} + {n^2}} }}} \right)^2} + {\left( {{{\alpha - 2\beta + \gamma } \over {\sqrt 6 }}} \right)^2} = 9$$<br><br>Locus is $${{{{(x + y + z)}^2}} \over 3} + {{{{(\ln - nz)}^2}} \over {{l^2} + {n^2}}} + {{{{(x - 2y + z)}^2}} \over 6} = 9$$<br><br>$${x^2}\left( {{1 \over 2} + {{{l^2}} \over {{l^2} + {n^2}}}} \right) + {y^2} + {z^2}\left( {{1 \over 2} + {{{n^2}} \over {{l^2} + {n^2}}}} \right) + 2zx\left( {{1 \over 2} - {{\ln } \over {{l^2} + {n^2}}}} \right) - 9 = 0$$<br><br>Since its given that ${x^2} + {y^2} + {z^2} = 9$<br><br>After solving l = n,<br><br>then, l $-$ n = 0